Question

If $A$ and $B$ are two square matrices of order $3\times 3$ which satisfy $AB=A$ and $BA=B$, then $(A+B{)}^7$ is equal to,

• A. $7(A+B)$
• B. $7.I_{3\times 3}$
• C. $64(A+B)$
• D. $128J$

Answer 1

The correct option is C $64(A+B)$

$AB=A⟹B=I$

$\Rightarrow BA=B⟹A=I$

$\Rightarrow A+B=2I$

$\Rightarrow {(A+B)}^7={\left(2I\right)}^7=128I$

$\Rightarrow {(A+B)}^7=64.2I$

$\Rightarrow {(A+B)}^7=64(A+B)$

Hence, the answer is $64(A+B).$